# Additivity and Chain Rules for Quantum Entropies via Multi-index Schatten Norms

## TL;DR

This paper explores how quantum entropies behave under tensor products and introduces new additivity and chain rules for quantum information tasks.

## Contribution

The paper introduces a general additivity statement for optimized sandwiched Rényi entropy of quantum channels using multi-index Schatten norms.

## Key findings

- A general additivity statement for optimized sandwiched Rényi entropy of quantum channels is established.
- Chain rules for Rényi conditional entropies are derived, similar to those in the generalized entropy accumulation theorem.
- The results strengthen the analysis of time-adaptive quantum cryptographic protocols.

## Abstract

The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of Devetak et al. (Commun Math Phys 266(1):37–63, 2006) to multi-index Schatten norms. As an application, we strengthen the additivity statement of Van Himbeeck and Brown (A tight and general finite-size security proof for quantum key distribution, 2025) thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of Metger et al. (Commun Math Phys 405(11):261, 2024).